All of the 4th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$2.00$ each for students, and the group paid $$33.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$13.00$ each for teachers and $$9.50$ each for students, and the group paid $$121.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+2y = 33}$ ${13x+9.5y = 121}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-13x-4y = -66}$ ${13x+9.5y = 121}$ Add the top and bottom equations together. $ 5.5y = 55 $ $ y = \dfrac{55}{5.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+2y = 33}$ to find $x$ ${6.5x + 2}{(10)}{= 33}$ $6.5x+20 = 33$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {13x+9.5y = 121}$ and get the same answer for $x$ ${13x + 9.5}{(10)}{= 121}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.